Orbital Estimation

By Dr. T.S. Kelso

May/June 1995

On the morning of 1995 February 9, I awoke a bit earlier than usual. Not that
I was any more anxious than usual to get to work that day. Instead, I realized
that I had a rare opportunity to watch a celestial event of historical import.
What made this event even more exciting was that I was able to predict just when
it would be visible from my new home in Montgomery, Alabama.

For several days, I had been watching the television coverage of the first
rendezvous of the US Space Shuttle with the Russian Mir Space Station. As
amazing as it was to watch these two large spacecraft maneuver around each other
on television, I was anxious to have the opportunity to watch this celestial
ballet live. After all, this is where "the rubber meets the road" for orbital
mechanics. With a fresh set of NORAD two-line orbital elements for both Mir and
the Space Shuttle and my trusty TrakStar prediction software using the NORAD
SGP4 orbital model, I began calculating when I might have my chance.

Of course, a number of conditions have to be met for you to be able to
visually observe an low-earth-orbiting satellite. First, the satellite must pass
over your location. That means that the satellite's orbit must have an
inclination greater than or (approximately) equal to your latitude (north or
south). For many Space Shuttle launches, the inclination is only 28.5 degrees
(the latitude of the Kennedy Space Center where the Space Shuttle is launched
from) and only a small percentage of the southern United States will even "see"
the Space Shuttle pop above the horizon. On this mission, however, the Space
Shuttle was launched into a 51.6-degree inclination in order to rendezvous with
the Mir Space Station. That meant that our first condition would be met for a
large percentage of the Earth's populated landmasses.

The next condition which must be met is for the satellite to pass over your
location at the right time of day. This condition is very important.
Basically, the observer must be in darkness and the satellite must be
illuminated by sunlight to be visible. For the observer to be in darkness, the
Sun must be six degrees or more below the horizon. For a satellite at the
altitude of Mir or the Space Shuttle, it must be passing near the Earth's
terminator (the line on the Earth's surface dividing day and night) within an
hour or so after the onset of nightfall or before daybreak. If the satellite
passes over at some other time of day, it simply will not be visible to the
naked eye.

As luck would have it, the conditions were met for this mission, although
they wouldn't come together for me until after the rendezvous was over.
Nonetheless, both spacecraft, with their crews of astronauts and cosmonauts,
would be chasing each other across the Alabama morning sky. Now, all that was
needed was good weather.

My calculations had shown that STS 63 would pop up out of the Earth's shadow
at 0554 CST at 14 degrees above the horizon, just north of west. Two minutes
later, the Mir Space Station would appear at the same location. Two minutes
might not seem like much of a separation, but at 7.5 km/sec, that's a distance
of almost 1,000 km. Both spacecraft would rise to just over 45 degrees above the
horizon before heading back to the horizon to the southeast. Once again, luck
was on my side since I had a nice unobstructed view in this direction.

As I walked outside, it was a brisk -3 degrees Celsius (okay, brisk for
Montgomery), but the sky was crystal clear. I started the car and began looking
for STS 63. I didn't have to wait long, with STS 63 popping into view exactly
when and where TrakStar had said it would. Suddenly, it didn't seem as cold any
more. I watched as it arced up toward the apex of its trajectory and then looked
back down to the horizon. Again, just as expected, there was the Mir space
station at the same place I'd first seen the Space Shuttle. Wow!

I was actually seeing both the Space Shuttle and the Mir Space Station in the
sky at the same time! And, they were right on schedule according to my program
(that always makes me feel a lot better). I couldn't have asked for a better
view, either. With both Venus and Jupiter in the morning sky, it was easy to use
them as benchmarks to determine the brightness of the two spacecraft—each
was about as bright as Jupiter and pretty tough to miss. It really was wonderful
to experience the excitement of calculating a visual satellite pass and watching
it unfold, as predicted. It sure brings those numbers to life.

Since this will not be the last time these two spacecraft will meet in space,
you will have the opportunity to go out and watch these events for yourself. In
fact, the Space Shuttle is scheduled to dock with Mir sometime later this year.
So, what do you need to do to prepare to observe?

As was pointed out in my previous columns, the first two things you'll need
will be current element sets and the appropriate orbital model. The most
widely-available source of orbital data for Mir is the NORAD two-line element
sets. These elements are available via Internet at archive.afit.af.mil in the
directory pub/space [no longer available]
and via dial-up modem on the Celestial BBS at (334) 409-9280 [dial-up BBS no
longer available] in the Satellite directory of the Files section. These
elements are updated every weekday (except holidays). The Mir elements can be
found in the file MIR.TLE. These elements are also echoed to many other Internet
sites, VANs (Value-Added Networks such as CompuServe and America Online), and
BBSs around the world. For the docking of the Space Shuttle to the Mir Space
Station later this year, your best bet to be prepared is to predict viewing
conditions using the Mir orbital elements until the elements for the Space
Shuttle become available (Mir has nowhere near the maneuvering capability of the
Space Shuttle).

Why are timely orbital elements important? For Mir, they are important
because the orbit is constantly changing due to the effects of changing
atmospheric drag and maneuvers to maintain the space station's orbit. Mir also
maneuvers to support rendezvous with the Soyuz and Progress spacecraft which
supply it and, now, the Space Shuttle. We'll see just how important this is in a
little bit.

For the orbital model, any package that implements the NORAD SGP4 model
should suffice. I use TrakStar because it does one thing many other tracking
programs do not: it calculates visible-only passes. That is, it can be set to
output data only when the satellite is illuminated and the observer is in
darkness. It doesn't output graphically, but the tabular ASCII output can be
easily imported into any spreadsheet software or other plotting package of your
choice to produce a trajectory. The key ingredient, however, is the use of the
SGP4 orbital model.

Here's why. In an ideal orbit, the basic orbital elements are constants. That
is, the orbital altitude, eccentricity, inclination, and orientation of the
orbit in space are all fixed. In real life, however, perturbations on the orbit
due to the nonuniform density of the Earth and atmospheric drag cause
fluctuations in a satellite's orbit. Let's look some specifics for Mir on the
day of my observation to illustrate these effects.

Assuming the mean elements of this element set to be constants (which many
non-SGP4 programs do), would yield a semi-major axis (half the distance between
the satellite's closest and farthest approaches to the Earth) of 6,769 km.
However, an examination of the semi-major axis over a time period of six hours
shows it to be far from constant, varying by about 12 km over a single orbit
(see Figure 1). And, while not immediately obvious from this plot, the altitude
is slowly decaying with time. At this altitude, an error of 1 km in altitude
corresponds to just over a 1 sec error per revolution in the orbital
period—that's about 20 seconds per day. Larger errors in altitude will
produce correspondingly larger errors in time for the position of the
satellite.

Figure 1. Semi-Major Axis (km) vs. Time (hr)

Looking next at the right ascension of the ascending node, the direction in
space where the orbit plane crosses the Earth's equator from south to north, we
see that this node precesses at a rate of over 4 degrees per day (see Figure 2).
At Mir's altitude, that can cause an error of about 500 km per day! Also, notice
that while there is a small periodic effect, the secular (trending) effect is
quite pronounced.

Figure 2. Right Ascension of the Ascending Node (deg) vs. Time (hr)

Finally, to give some idea of the complexity of the fluctuations that can be
seen in the orbital elements, let's look at how the argument of perigee for Mir
changes over a six-hour period (Figure 3). The argument of perigee is the angle
between the right ascension of the ascending node and the direction of the
perigee, or closest approach to the Earth's center, measured along the orbit
path. As this angle fluctuates, the point where the satellite is lowest also
fluctuates, resulting in even more errors. In our example, several periodic
effects appear to be present.

Figure 3. Argument of Perigee (deg) vs. Time (hr)

When we combine all the effects in the SGP4 model and compare it to a simple
two-body propagator, we find an error of 150 km after only six hours and almost
400 km by the end of the day! Given element sets which may be several days old,
the errors can be on the order of thousands of kilometers! And using a model
which incorporates the perturbations in a manner contradictory to SGP4 can make
the errors even worse.

In our next column, we'll look at what criteria are used to determine when an
element set needs updating and what mathematical methods are used. Once we're
done with that, we'll start exploring some of the basics of taking the output of
SGP4 to produce calculations like those in TrakStar. See you next time!